Uncertainty quantiﬁcation works relevant to ﬁssion yields and decay data

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In the present paper, ﬁrstly, we review our previous works on uncertainty quantiﬁcation (UQ) of reactor physics parameters. This consists of (1) development of numerical tools based on the depletion perturbation theory (DPT), (2) linearity of reactor physics parameters to nuclear data, (3) UQ of decay heat and its reduction, and (4) correlation between decay heat and b-delayed neutrons emission.

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EPJ Nuclear Sci. Technol. 4, 43 (2018) Nuclear Sciences © G. Chiba and S. Nihira, published by EDP Sciences, 2018 & Technologies https://doi.org/10.1051/epjn/2018022 Available online at: https://www.epj-n.org REGULAR ARTICLE Uncertainty quantiﬁcation works relevant to ﬁssion yields and decay data Go Chiba* and Shunsuke Nihira Hokkaido University, Kita 13 Nishi 8, Kita-ku, Sapporo 060-8628, Japan Received: 29 September 2017 / Received in ﬁnal form: 21 January 2018 / Accepted: 14 May 2018 Abstract. In the present paper, ﬁrstly, we review our previous works on uncertainty quantiﬁcation (UQ) of reactor physics parameters. This consists of (1) development of numerical tools based on the depletion perturbation theory (DPT), (2) linearity of reactor physics parameters to nuclear data, (3) UQ of decay heat and its reduction, and (4) correlation between decay heat and b-delayed neutrons emission. Secondly, we show results of extensive calculations about UQ on decay heat with several different numerical conditions by the DPT- based capability of a reactor physics code system CBZ. 1 Introduction 2.1 Development of numerical tools based on DPT The adjoint-based procedure requires sensitivity proﬁles of Generally there are two numerical procedures for uncertain- reactor physics parameters with respect to nuclear data, ty quantiﬁcation (UQ) of reactor physics parameters and these can be efﬁciently calculated by the perturbation induced by nuclear data uncertainty: the adjoint-based or generalized perturbation theory. If reactor physics procedure and the stochastic-based procedure. In recent parameters are quantities obtained from nuclides depletion years, the stochastic-based procedure has been adopted problems, DPT is generally utilized. DPT has been well frequently by virtue of drastic advancements of computer established in the past [1,2], but there are no computer resources, but the adjoint-based procedure is still powerful codes which implement capability of sensitivity calcula- because it can yield sensitivity proﬁles of reactor physics tions based on DPT at present. Theoretical description of parameters with respect to nuclear data without any DPT is omitted in the present paper, but interested readers statistical ﬂuctuations with short computation time relative can see it in the other papers [1–4]. We have developed our to the stochastic-based procedure. Sensitivity proﬁles of own reactor physics code system CBZ and have imple- reactor physics parameters to nuclear data are quite mented DPT-based sensitivity calculation capability into beneﬁcial quantities to understand physical mechanism it. On light water reactor analyses, this capability had been behind numerical results, and to identify important nuclear limited to single pincell problems originally [3], but it has data for accurate estimations of reactor physics parameters. been extended to multi-cell problems [4]. More recently, Our research group in Hokkaido University has carried DPT has been extended for depletion calculations with the out several works about UQ based on the adjoint-based predictor–corrector method, which is mandatory in procedure. We have focused mainly on nuclide transmuta- calculations of actual fuel assemblies including burnable tion problems such as nuclear fuel depletion and b-delayed neutron absorbers [5]. neutrons emissions, and have developed numerical tools based on the DPT. The present paper consists of two parts; 2.2 On linearity of reactor physics parameters to the ﬁrst part is to review our previous works on UQ based on nuclear data DPT, and the second part provides some numerical results of UQ for decay heat in various conditions using our tools. Although the adjoint-based procedure possess several advantages, it introduces an important assumption: linear 2 Reviews of UQ works done at Hokkaido dependence of reactor physics parameters with respect to University nuclear data. This assumption should be valid if nuclear data uncertainties are small, but it is not the case if these In the present section, our previous works on UQ are brieﬂy uncertainties are large. Since probability density functions reviewed. of nuclear data are not explicitly deﬁned in evaluated nuclear data ﬁles, the normal distributions are generally * e-mail: go_chiba@eng.hokudai.ac.jp assumed. If linearity of reactor physics parameters to This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
2 G. Chiba and S. Nihira: EPJ Nuclear Sci. Technol. 4, 43 (2018) Fig. 1. Example of distorted probability density distribution. Fig. 3. Skewness and kurtosis of number densities after depletion of several nuclides. In most nuclides, linearity of number densities after depletion with respect to nuclear data has been conﬁrmed, but some exceptional cases have been observed. Skewness and kurtosis calculated are shown in Figure 3. Error bars are statistical uncertainties estimated by the bootstrap method [7]. It is notable that non-negligible difference in standard deviations between the adjoint-based procedure and the stochastic-based procedure are observed in these number densities because of non-linear effect which can be taken into account only by the stochastic-based procedure. Large skewness of some europium and gadolinium isotopes are caused by large uncertainty of half life of samarium- 151. Since JENDL/FPD-2011 gives no uncertainty to this nuclear data, we have assumed 100% uncertainty. This large uncertainty results in non-linear effect to its daughter Fig. 2. Relationship between skewness and probability over 2s. nuclides generation, and distorted distributions of these nuclides number densities are obtained. It is interesting to point out that if uncertainty is evaluated not to half life but to decay constant, this should not occur because generation nuclear data holds, probability density distribution of of daughter nuclide can be well approximated by the ﬁrst- reactor physics parameters should be also the normal order as distribution. On the other hand, if the linearity does not hold, the probability density distribution should be N 2 ðtÞ ¼ 1 N 1 ð0Þexpðl1 tÞ ≈ N 1 l1 t; ð1Þ distorted. Figure 1 shows an example of distorted distribution. In this case, probability that a statistical where Ni(t) is number density of nuclide i at t and li is parameter takes value larger than its average plus 2s is decay constant of nuclide i. It is also interesting to point out around 4.3%, which is around 2.3% in case of the normal that this might be totally different if we consider number distribution. This might be problematic in parameter densities of nuclides which are in an equilibrium state as speciﬁcations with proper margin in nuclear reactor safety suggested by Endo [8]. Number density of such a nuclide analyses. Figure 2 shows relationship between skewness of can be represented as a probability density distribution and probability that a parameter takes larger value than its average plus 2s. This R NðtÞ ≈ ¼ RT 1=2 =ðln2Þ; ð2Þ ﬁgure is prepared by assuming second-order dependence of l an output parameter to an input parameter of the normal distribution. where T1/2 is half life. This suggests that number density In our previous study, a linearity check has been done should be linear on its half life. for ﬁssion products nuclides generation of light water reactors by using the stochastic-based procedure [6]. This 2.3 Uncertainty quantiﬁcation of decay heat and its has been accomplished by observing high-order statistical reduction by nuclear data adjustment quantities, such as skewness and kurtosis, and deviations from values of the normal distribution are regarded as Nuclear data-induced uncertainty of decay heat have been indices to check the linearity. quantiﬁed with the adjoint-based procedure. In addition,
G. Chiba and S. Nihira: EPJ Nuclear Sci. Technol. 4, 43 (2018) 3 Fig. 5. Correlation matrix of decay heat and b-delayed neutron activity after uranium-235 ﬁssion by thermal neutrons. heat (from 32 to 62) and (3) b-delayed neutron activities (from 63 to 94). This result suggests that the correlation among these blocks is insigniﬁcant, and that independent treatments of decay heat and b-delayed neutrons emissions are possible. 3 UQ of decay heat in various conditions with DPT Fig. 4. Nuclear data-induced uncertainty of decay heat. The upper one is the short-term decay heat and the lower is the long- By virtue of DPT, we carry out extensive calculations term one. about UQ on decay heat with the code system CBZ. In these calculations, all the ﬁssion product nuclides given in evaluated nuclear data ﬁles are explicitly treated. Correla- uncertainty reduction using the nuclear data adjustment tion in ﬁssion yields data among different nuclides is taken method with measurement data has been attempted by our into account by a method proposed by Katakura [12]. research group [9]. We have quantiﬁed uncertainties of Figure 6 shows nuclear data-induced uncertainty of decay both the short-term decay heat, which is important in heat after uranium-235 ﬁssions with thermal neutrons with safety analyses of nuclear reactors, and the long-term decay different irradiation periods. Decay heat uncertainty after heat, which is important in management of spent nuclear ﬁssion burst is almost the same as that after 1-second fuels. Figure 4 shows uncertainties of short-term and long- irradiation, so it is not shown in the present paper. term decay heat before and after the nuclear data JENDL/FPY-2011 and JENDL/FPD-2011 are used. In adjustment. We have demonstrated that the uncertainties these calculations, 100% uncertainty is assumed to nuclear of the short-term and long-term decay heat can be reduced data to which uncertainty is not evaluated in these ﬁles. It by using measurement data of decay heat after ﬁssion pulse can be observed that decay energy uncertainty is dominant and that of post irradiation examination, respectively. in all the irradiation periods. Next, decay heat uncertainty is calculated with an assumption that 0% uncertainty is 2.4 On correlation between decay heat and b-delayed given to nuclear data which have no uncertainty informa- neutrons emission tion. Results are shown in Figure 7. The uncertainty is signiﬁcantly reduced from that shown in Figure 6, so it is b-delayed neutrons emission is quite important in nuclear preferred that uncertainty of decay energy is evaluated reactors kinetics. We have quantiﬁed nuclear data-induced somehow in the JENDL library. The same tendency is uncertainty of b-delayed neutron emission rates in the observed when the ENDF and JEFF libraries are used, but frame of summation calculations [10]. In addition, since it is less signiﬁcant than the JENDL case. nuclear data relevant to the b-delayed neutrons emission Finally decay heat uncertainty is calculated for fuel are also important in decay heat calculations, we have pincell model of typical light water reactors. Uranium-235 examined correlation between nuclear data-induced un- enrichment of this fuel is 4.1 wt.%. Decay heat uncertainty certainty of decay heat and that of b-delayed neutrons is calculated with different burnup: 1, 5, 10, 20 and emission [11]. Correlation matrix of decay heat and 40 GWD/t. Figure 8 shows decay heat uncertainty with b-delayed neutron activity after uranium-235 ﬁssion by several burnup. It is interesting to point out that the decay thermal neutrons is shown in Figure 5. This matrix is heat uncertainty is not signiﬁcantly dependent on burnup, decomposed into three blocks: g component of decay heat and that it is quite similar with that of ﬁnite-period (from 1 to 31 in x and y axes), (2) b component of decay irradiation case shown in Figure 7.
4 G. Chiba and S. Nihira: EPJ Nuclear Sci. Technol. 4, 43 (2018) Fig. 6. Nuclear data-induced uncertainty of decay heat after uranium-235 ﬁssion with thermal neutrons with different irradiation periods. Default uncertainty is assumed 100%. Figure 9 shows a correlation matrix of decay heat y axes) and 1 GWD/t burnup (from 27 to 52). In each sub- uncertainties after 100 000-second irradiation and 1 GWD/t matrix, smaller index corresponds to short cooling time. burnup. This matrix is decomposed into two blocks: decay Strong positive correlation can be observed between these heat after 100 000-second irradiation (from 1 to 26 in x and two different conditions.
G. Chiba and S. Nihira: EPJ Nuclear Sci. Technol. 4, 43 (2018) 5 Fig. 7. Nuclear data-induced uncertainty of decay heat after uranium-235 ﬁssion with thermal neutrons with different irradiation periods. Default uncertainty is assumed 0%.
6 G. Chiba and S. Nihira: EPJ Nuclear Sci. Technol. 4, 43 (2018) Fig. 8. Nuclear data-induced uncertainty of decay heat after uranium-235 ﬁssion with thermal neutrons with different burnup. Default uncertainty is assumed 0%.
G. Chiba and S. Nihira: EPJ Nuclear Sci. Technol. 4, 43 (2018) 7 heat with several different numerical conditions by the DPT-based capability of the reactor physics code system CBZ. References 1. A. Gandini, Nucl. Sci. Eng. 38, 38 (1969) 2. M.L. Williams, Nucl. Sci. Eng. 70, 20 (1979) 3. G. Chiba, M. Tsuji, T. Narabayashi, J. Nucl. Sci. Technol. 50, 751 (2013) 4. G. Chiba, Y. Kawamoto, T. Narabayashi, Ann. Nucl. Energy 96, 313 (2016) 5. G. Chiba, J. Nucl. Sci. Technol. 55, 450 (2018) Fig. 9. Correlation matrix of decay heat uncertainties after 6. S. Nihira, G. Chiba, in Proceeding of Reactor Physics Asia 100 000-second irradiation and 1 GWD/t burnup. Conference 2017 (RPHA2017), Chengdu, China, August 24– 25, 2017 7. T. Endo, T. Watanabe, A. Yamamoto, J. Nucl. Sci. Technol. 52, 993 (2015) 8. T. Endo, private communication 4 Conclusion 9. Y. Kawamoto, G. Chiba, J. Nucl. Sci. Technol. 54, 213 (2017) In the present paper, we have reviewed our previous works 10. G. Chiba, Ann. Nucl. Energy 85, 846 (2015) on UQ of reactor physics parameters. Furthermore, we 11. G. Chiba, Ann. Nucl. Energy 101, 23 (2017) have carried out extensive calculations about UQ on decay 12. J. Katakura, J. Nucl. Sci. Technol. 50, 799 (2016) Cite this article as: Go Chiba, Shunsuke Nihira, Uncertainty quantiﬁcation works relevant to ﬁssion yields and decay data, EPJ Nuclear Sci. Technol. 4, 43 (2018)